I didn't see the Fox segment (mostly because, God help me, why would I watch Fox News), so I don't know what their argument was. The TPM post assumes that question is total bunk, but it might not be. I think it's just not worded very well. Is lowering the minimum wage better for all workers? No. Is it better for some? Probably yes. As with any public policy, some people are usually made better off and some worse off.
Here's a back-of-the-envelope example: A business with 10 full-time employees pays them $5 an hour. Their annual payroll (not including any benefits) is approx: $96,000. If the minimum wage goes from $5 to $7 an hour, their annual payroll shoots to $134,400. To a small business owner, this can be very burdensome. So what would an employer do? Well, chances are employees may have to go, they may have to lose hours and work part-time, or maybe the business can't hire new people and will work with less.
Who is made better off by a minimum wage? Worker's who keep their jobs or who are able to find jobs. Who is worse off? Employer's on the margin, workers who lose their jobs or lose hours, and workers who want to work but can't find jobs. Remember that a minimum wage generally only affects those with very-low human capital (training, education, job skills, etc). So it's only affecting a certain group of people.
This is not really an argument against the minimum wage, but one to say that it's not crazy to ask how the minimum wage is affecting employment during a recession, because it might be. One problem is that we tend to want the policy to reflect a living wage, not a minimum wage.
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